CW complexes and manifolds
What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
Solution 1:
Smooth closed manifolds can always be given a CW structure via Morse functions, but one has to be careful in constructing the attaching maps (for instance, one must choose a regular neighborhood of the $k$-skeleton and perturb the gradient flow to define attaching maps for the $(k+1)$-cells). For more details, see the answers to this MathOverflow question, especially Ryan Budney's.
Closed topological manifolds of dimension other than four are homeomorphic to CW complexes. The reference is Kirby-Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull AMS 75 (1969). My reference for this is (again) Ryan Budney's answer to this MathOverflow question. The comments to the answer and to this MO question indicate that Kirby-Siebenmann prove that every compact (top) manifold of dimension other than $4$ is indeed a CW complex, but the question is still open in dimension 4. (Another comment points to these slides by A. Ranicki.)
I'm not sure how the classification proceeds for open manifolds.
Solution 2:
This is an addendum to Neal's answer (concerning the 4-dimensional case):
Every connected noncompact 4-manifold admits a PL structure, hence, a triangulation (F.Quinn, "Ends of maps-III", 1982).
For compact 4-manifolds the problem of existence of a CW-complex (as well as a handlebody) structure is open, see Question 1.3 in this list of problems.